A model of computer memory Ben Fairbairn Birkbeck, University of - - PowerPoint PPT Presentation

A model of computer memory

Ben Fairbairn Birkbeck, University of London Groups St Andrews, August 2013

Joint work with Peter Cameron and Max Gadouleau

An application of group theory to genetics

Ben Fairbairn Birkbeck, University of London Groups St Andrews, August 2013

The basic problem

GAP, Version 4.6.4 of 04-May-2013 (free software, GPL) http://www.gap-system.org Architecture: i686-pc-cygwin-gcc-default32 Libs used: gmp, readline . . . Error, exceeded the permitted memory (‘-o’ command line option)

Swapping two variables: undergraduate method

x y t action 1 2

Swapping two variables: undergraduate method

x y t action 1 2

• 1

2 1 x ← t

Swapping two variables: undergraduate method

x y t action 1 2

• 1

2 1 x ← t 2 2 1 y ← x

Swapping two variables: undergraduate method

x y t action 1 2

• 1

2 1 x ← t 2 2 1 y ← x 2 1 1 t ← y

Swapping two variables: the “XOR” trick

x y action 1 2

Swapping two variables: the “XOR” trick

x y action 1 2

• 3

2 x + y ← x

Swapping two variables: the “XOR” trick

x y action 1 2

• 3

2 x + y ← x 3 1 x − y ← y

Swapping two variables: the “XOR” trick

x y action 1 2

• 3

2 x + y ← x 3 1 x − y ← y 2 1 x − y ← x

Main definitions (Gadouleau & Riis)

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer.

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer. ◮ A bijection f : An → An is an instruction if

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer. ◮ A bijection f : An → An is an instruction if for

(x1, . . . , xn) ∈ An such that f : (x1, . . . , xn) → (y1, . . . , yn)

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer. ◮ A bijection f : An → An is an instruction if for

(x1, . . . , xn) ∈ An such that f : (x1, . . . , xn) → (y1, . . . , yn) we have that xj = yj for all j = i for some 1 ≤ i ≤ j.

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer. ◮ A bijection f : An → An is an instruction if for

(x1, . . . , xn) ∈ An such that f : (x1, . . . , xn) → (y1, . . . , yn) we have that xj = yj for all j = i for some 1 ≤ i ≤ j.

◮ We say that f updates the ith register.

Main definitions (Gadouleau & Riis)

◮ Let A = {0, . . . , q − 1} be a set and let n be a positive integer. ◮ A bijection f : An → An is an instruction if for

(x1, . . . , xn) ∈ An such that f : (x1, . . . , xn) → (y1, . . . , yn) we have that xj = yj for all j = i for some 1 ≤ i ≤ j.

◮ We say that f updates the ith register. ◮ We call elements of An states.

An example

If A = {0, 1} (so q = 2) and n = 4 then An consists of. . .

An example

If A = {0, 1} (so q = 2) and n = 4 then An consists of. . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

◮ S. Burckel “Closed Iterative Calculus” Theoretical Computer

Science, 158, 1996, 371–378

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for Graph Entropy and Network Coding Based Communications” IEEE Transactions on Information Theory,

• Vol. 57, No. 10, 2011, 6703–6717

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Demongeot, M. Kaufman, A. Naldi, E.H. Snoussi and D. Thieffry “On circuit functionality in Boolean networks” Bulletin of Mathematical Biology, in press 2013

◮ S. Kauffman “Metabolic Stability and Epigenesis in Randomly

Connected Nets” Journal of Theoretical Biology, 22, 1969, 437–467

◮ ´

E Remy, P. Ruet and D. Thieffry “Graphic requirements for multistability and attractive cycles in a Boolean Dynamical Framework” Advances in Applied Mathematics, 41, 2008, 335–350

◮ R. Thomas “Boolean formalization of genetic control circuits”

Journal of Theoretical Biology, 42(3), 563–585, 1973

Questions

Note that each of the Coxeter generators are instructions.

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)?

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

◮ Can we generate Sym(An) with n instructions?

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

◮ Can we generate Sym(An) with n instructions? ◮ What about Alt(An)?

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

◮ Can we generate Sym(An) with n instructions? ◮ What about Alt(An)?

If q is a prime power then we can view An as a finite vector space and restrict our attention to instructions that are linear maps.

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

◮ Can we generate Sym(An) with n instructions? ◮ What about Alt(An)?

If q is a prime power then we can view An as a finite vector space and restrict our attention to instructions that are linear maps.

◮ Can we generate GLn(q) with n instructions?

Questions

Note that each of the Coxeter generators are instructions.

◮ Is this the smallest set of instructions we need to generate

Sym(An)? We need at least one for each register, so we should really ask. . .

◮ Can we generate Sym(An) with n instructions? ◮ What about Alt(An)?

If q is a prime power then we can view An as a finite vector space and restrict our attention to instructions that are linear maps.

◮ Can we generate GLn(q) with n instructions? ◮ What about SLn(q)?

Some results

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

Some results

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

Theorem (Cameron, F., Gadouleau, 201?)

Unless q = 2 and n = 2 or 3, the group Alt(An) can be generated by n instructions.

Some results

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

Theorem (Cameron, F., Gadouleau, 201?)

Unless q = 2 and n = 2 or 3, the group Alt(An) can be generated by n instructions.

Theorem (Cameron, F., Gadouleau, 201?)

dito GLn(q) and SLn(q).

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

◮ |A| = 2;

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

◮ |A| = 2; ◮ |A| > 2 and n is odd;

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

◮ |A| = 2; ◮ |A| > 2 and n is odd; ◮ |A| > 2 and n is even;

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

◮ |A| = 2; ◮ |A| > 2 and n is odd; ◮ |A| > 2 and n is even;

In each case we write down explicit permutations that generate a 2-transitive (and thus primitive) group

Sketch proof for Sym(An)

Theorem (Cameron, F., Gadouleau, 201?)

Unless n = q = 2, the group Sym(An) can be generated by n instructions.

“Proof”

It’s convenient to consider different cases separately:

◮ |A| = 2; ◮ |A| > 2 and n is odd; ◮ |A| > 2 and n is even;

In each case we write down explicit permutations that generate a 2-transitive (and thus primitive) group and apply Jordan’s Theorem (our permutations are chosen so that some power of one

• f them is a 3-cycle).

Further results

Further results

◮ Every element of Sym(An) is a word of instructions of length

at most 2n − 1

Further results

◮ Every element of Sym(An) is a word of instructions of length

at most 2n − 1 and we know the extreme case: maximal Hamming distance.

Further results

◮ Every element of Sym(An) is a word of instructions of length

at most 2n − 1 and we know the extreme case: maximal Hamming distance.

◮ Every element of GLn(q) is a word of instructions of length at

most ⌊3n/2⌋

Further results

◮ Every element of Sym(An) is a word of instructions of length

at most 2n − 1 and we know the extreme case: maximal Hamming distance.

◮ Every element of GLn(q) is a word of instructions of length at

most ⌊3n/2⌋ and we know the extreme case if q = 2 and n is even: derangement permutation matrices of order 2.

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set?

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set? NO!

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set? NO! If we uniformly at random choose an instruction for each register, does the probability that we obtain a generating set for Sym(An) tend to 1 as |An| → ∞?

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set? NO! If we uniformly at random choose an instruction for each register, does the probability that we obtain a generating set for Sym(An) tend to 1 as |An| → ∞?

◮ Given an instruction, how do we find instructions updating the

• ther registers that combined give us a minimal generating

set?

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set? NO! If we uniformly at random choose an instruction for each register, does the probability that we obtain a generating set for Sym(An) tend to 1 as |An| → ∞?

◮ Given an instruction, how do we find instructions updating the

• ther registers that combined give us a minimal generating

set?

◮ Given a minimal generating set, how can we find a program

(list of instructions) that equal a given element of Sym(An)?

Questions, open problems, conjectures, jokes, songs, limericks, cake recipes, . . .

◮ Can all instructions be incorporated into a minimal generating

set? NO! If we uniformly at random choose an instruction for each register, does the probability that we obtain a generating set for Sym(An) tend to 1 as |An| → ∞?

◮ Given an instruction, how do we find instructions updating the

• ther registers that combined give us a minimal generating

set?

◮ Given a minimal generating set, how can we find a program

(list of instructions) that equal a given element of Sym(An)?

◮ All of the above with Sym(An) replaced by Alt(An), GLn(q),

SLn(q), etc?

. . . stories, anecdotes, baking tips, poetry, . . .

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains.

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains. Which groups are internally computable?

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains. Which groups are internally computable? (All groups discussed so far are, various other matrix groups in their natural representations are not.)

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains. Which groups are internally computable? (All groups discussed so far are, various other matrix groups in their natural representations are not.)

◮ An internally computable group H < G ≤ Sym(An) is fast in

G if the shortest program expressing a given element of H consists of instructions from H.

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains. Which groups are internally computable? (All groups discussed so far are, various other matrix groups in their natural representations are not.)

◮ An internally computable group H < G ≤ Sym(An) is fast in

G if the shortest program expressing a given element of H consists of instructions from H. Which groups are fast?

. . . stories, anecdotes, baking tips, poetry, . . .

◮ A permutation group is internally computable if it can be

generated by the instructions it contains. Which groups are internally computable? (All groups discussed so far are, various other matrix groups in their natural representations are not.)

◮ An internally computable group H < G ≤ Sym(An) is fast in

G if the shortest program expressing a given element of H consists of instructions from H. Which groups are fast? (Any internally computable group is fast in itself, Alt(An) is not fast in Sym(An) (3-cycles) similarly SLn(q) is not fast in GLn(q).)

Cheers for listening!